Apple’s M1 processor and the full 128-bit integer product

If I multiply two 64-bit integers (having values in [0, 2⁶⁴)), the product requires 128 bits. Intel and AMD processors (x64) can compute the full (128-bit) product of two 64-bit integers using a single instruction (mul). ARM processors, such as those found in your mobile phone, require two instructions to achieve the same result: mul computes the least significant 64 bits while mulh computes the most significant 64 bits.

I believe that it has often meant that computing the full 128-bit product was more expensive, everything else being equal, on ARM processors than on x64 (Intel) processors. However, the instruction set does not have to determine the performance. For example, ARM processors can recognize that I am calling both instructions (mul  and mulh) and process them more efficiently.

To explore the problem, let us pick two pseudo-random number generators, splitmix and wyhash:

uint64_t splitmix() {
    uint64_t z = (state += UINT64_C(0x9E3779B97F4A7C15));
    z = (z ^ (z >> 30)) * UINT64_C(0xBF58476D1CE4E5B9);
    z = (z ^ (z >> 27)) * UINT64_C(0x94D049BB133111EB);
    return z ^ (z >> 31);
}

uint64_t wyhash() {
    state += 0x60bee2bee120fc15ull;
    __uint128_t tmp = (__uint128_t)(state)*0xa3b195354a39b70dull;
    uint64_t m1 = (tmp >> 64) ^ tmp;
    tmp = (__uint128_t)m1 * 0x1b03738712fad5c9ull;
    return (tmp >> 64) ^ tmp;
 }

As I reported earlier, wyhash should almost always be faster on an Intel or AMD processor as it is only two multiplications with an addition whereas the splitmix function is made of two multiplications with several other operations. However, wyhash requires two full multiplications whereas splitmix requires only two 64-bit products. If the two full multiplications in wyhash are equivalent two four multiplications, then wyhash becomes more expensive.

I wrote a small C++ benchmark to measure the time (in nanoseconds) that it takes to compute a random value using Apple’s new M1 processor (ARM). The compiler is Apple clang version 12 which comes by default on the new Apple Silicon laptops.

	Apple M1
wyhash	0.30 ns/value
splitmix	0.45 ns/value

The wyhash generator is much faster than splitmix on the Apple M1 processor (by about 50%) which suggests that Apple Silicon is efficient at computing the full 128-bit product of two 64-bit integers.

Credit: Maynard Handley suggested this blog post.

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